Abstract
The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.
Highlights
In this paper 1, Soergel gave a combinatorial description of a certain category of HarishChandra bimodules over a simple Lie algebra g
Rouquier 22, 23 defined an action of the Coxeter braid group associated to W on the category of complexes of Soergel bimodules up to homotopies, which is related to a braid group action using Harish-Chandra bimodules that had been known for some time
These complexes were later used in an alternative construction of a triply graded link homology theory categorifying the HOMFLY-PT polynomial 25–28
Summary
In this paper 1 , Soergel gave a combinatorial description of a certain category of HarishChandra bimodules over a simple Lie algebra g. In the categorifications 9, 10 of quantum groups, 2 morphisms are given by linear combinations of planar diagrams, modulo local relations. Rouquier 22, defined an action of the Coxeter braid group associated to W on the category of complexes of Soergel bimodules up to homotopies, which is related to a braid group action using Harish-Chandra bimodules that had been known for some time These complexes were later used in an alternative construction of a triply graded link homology theory categorifying the HOMFLY-PT polynomial 25–28. The planar diagrams of our paper are two-dimensional encodings of these foams, essentially projections of the foams onto the yz-plane along the x-axis It was shown in 32 that the action of the braid group on the homotopy category of Soergel bimodules extends to a projective action of the category of braid cobordisms. Additional statements relating this paper to either newer papers or to previous versions of this paper are found sparsely under a similar “Addendum” heading
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